Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$.
Answer: Since $x^2 + x + 1$ is a factor of $(x^2 + x + 1)(x - 1) = x^3 - 1,$ and so also a factor of $x(x^3 - 1) = x^4 - x,$ the remainder when $(x^4 - 1)(x^2 - 1)$ is divided by $x^2 + x + 1$ is the same as the remainder of
\[(x - 1)(x^2 - 1) = x^3 - x^2 - x + 1.\]This has the same remainder when $1 - x^2 - x + 1 = -x^2 - x + 2 = -(x^2 + x + 1) + 3$ is divided by $x^2 + x + 1,$ which is $\boxed{3}.$